Diagrams.ThreeD.Shapes:frustum from diagrams-lib-1.3.0.3, A

Average Error: 6.4 → 0.7

Time: 7.5s

Precision: binary64

Math

\[2 \cdot \left(\left(x \cdot y + z \cdot t\right) - \left(\left(a + b \cdot c\right) \cdot c\right) \cdot i\right) \]

↓

\[\begin{array}{l} t_1 := \left(c \cdot \left(a + b \cdot c\right)\right) \cdot i\\ \mathbf{if}\;t_1 \leq -\infty:\\ \;\;\;\;2 \cdot \mathsf{fma}\left(y, x, \mathsf{fma}\left(c \cdot \left(b \cdot i\right) + a \cdot i, -c, t \cdot z\right)\right)\\ \mathbf{elif}\;t_1 \leq 5 \cdot 10^{+302}:\\ \;\;\;\;2 \cdot \mathsf{fma}\left(y, x, \mathsf{fma}\left(i, c \cdot \left(-\mathsf{fma}\left(c, b, a\right)\right), t \cdot z\right)\right)\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \mathsf{fma}\left(y, x, \mathsf{fma}\left(\mathsf{fma}\left(c \cdot i, b, a \cdot i\right), -c, t \cdot z\right)\right)\\ \end{array} \]

FPCore

(FPCore (x y z t a b c i)
 :precision binary64
 (* 2.0 (- (+ (* x y) (* z t)) (* (* (+ a (* b c)) c) i))))

↓

(FPCore (x y z t a b c i)
 :precision binary64
 (let* ((t_1 (* (* c (+ a (* b c))) i)))
   (if (<= t_1 (- INFINITY))
     (* 2.0 (fma y x (fma (+ (* c (* b i)) (* a i)) (- c) (* t z))))
     (if (<= t_1 5e+302)
       (* 2.0 (fma y x (fma i (* c (- (fma c b a))) (* t z))))
       (* 2.0 (fma y x (fma (fma (* c i) b (* a i)) (- c) (* t z))))))))

C

double code(double x, double y, double z, double t, double a, double b, double c, double i) {
	return 2.0 * (((x * y) + (z * t)) - (((a + (b * c)) * c) * i));
}

↓

double code(double x, double y, double z, double t, double a, double b, double c, double i) {
	double t_1 = (c * (a + (b * c))) * i;
	double tmp;
	if (t_1 <= -((double) INFINITY)) {
		tmp = 2.0 * fma(y, x, fma(((c * (b * i)) + (a * i)), -c, (t * z)));
	} else if (t_1 <= 5e+302) {
		tmp = 2.0 * fma(y, x, fma(i, (c * -fma(c, b, a)), (t * z)));
	} else {
		tmp = 2.0 * fma(y, x, fma(fma((c * i), b, (a * i)), -c, (t * z)));
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b, c, i)
	return Float64(2.0 * Float64(Float64(Float64(x * y) + Float64(z * t)) - Float64(Float64(Float64(a + Float64(b * c)) * c) * i)))
end

↓

function code(x, y, z, t, a, b, c, i)
	t_1 = Float64(Float64(c * Float64(a + Float64(b * c))) * i)
	tmp = 0.0
	if (t_1 <= Float64(-Inf))
		tmp = Float64(2.0 * fma(y, x, fma(Float64(Float64(c * Float64(b * i)) + Float64(a * i)), Float64(-c), Float64(t * z))));
	elseif (t_1 <= 5e+302)
		tmp = Float64(2.0 * fma(y, x, fma(i, Float64(c * Float64(-fma(c, b, a))), Float64(t * z))));
	else
		tmp = Float64(2.0 * fma(y, x, fma(fma(Float64(c * i), b, Float64(a * i)), Float64(-c), Float64(t * z))));
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_, c_, i_] := N[(2.0 * N[(N[(N[(x * y), $MachinePrecision] + N[(z * t), $MachinePrecision]), $MachinePrecision] - N[(N[(N[(a + N[(b * c), $MachinePrecision]), $MachinePrecision] * c), $MachinePrecision] * i), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

↓

code[x_, y_, z_, t_, a_, b_, c_, i_] := Block[{t$95$1 = N[(N[(c * N[(a + N[(b * c), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * i), $MachinePrecision]}, If[LessEqual[t$95$1, (-Infinity)], N[(2.0 * N[(y * x + N[(N[(N[(c * N[(b * i), $MachinePrecision]), $MachinePrecision] + N[(a * i), $MachinePrecision]), $MachinePrecision] * (-c) + N[(t * z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, 5e+302], N[(2.0 * N[(y * x + N[(i * N[(c * (-N[(c * b + a), $MachinePrecision])), $MachinePrecision] + N[(t * z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(2.0 * N[(y * x + N[(N[(N[(c * i), $MachinePrecision] * b + N[(a * i), $MachinePrecision]), $MachinePrecision] * (-c) + N[(t * z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Target

Original	6.4
Target	1.7
Herbie	0.7

\[2 \cdot \left(\left(x \cdot y + z \cdot t\right) - \left(a + b \cdot c\right) \cdot \left(c \cdot i\right)\right) \]

Derivation

1. Split input into 3 regimes

2. if (*.f64 (*.f64 (+.f64 a (*.f64 b c)) c) i) < -inf.0

   1. Initial program 64.0

      \[2 \cdot \left(\left(x \cdot y + z \cdot t\right) - \left(\left(a + b \cdot c\right) \cdot c\right) \cdot i\right) \]

   2. Simplified 9.4

      \[\leadsto \color{blue}{2 \cdot \mathsf{fma}\left(z, t, \mathsf{fma}\left(\mathsf{fma}\left(b, c, a\right), i \cdot \left(-c\right), x \cdot y\right)\right)} \]

   3. Taylor expanded in z around 0 9.4

      \[\leadsto 2 \cdot \color{blue}{\left(y \cdot x + \left(t \cdot z + -1 \cdot \left(c \cdot \left(i \cdot \left(c \cdot b + a\right)\right)\right)\right)\right)} \]

   4. Simplified 9.4

      \[\leadsto 2 \cdot \color{blue}{\mathsf{fma}\left(y, x, \mathsf{fma}\left(i \cdot \mathsf{fma}\left(c, b, a\right), -c, t \cdot z\right)\right)} \]

   5. Taylor expanded in c around 0 5.2

      \[\leadsto 2 \cdot \mathsf{fma}\left(y, x, \mathsf{fma}\left(\color{blue}{c \cdot \left(i \cdot b\right) + i \cdot a}, -c, t \cdot z\right)\right) \]

   if -inf.0 < (*.f64 (*.f64 (+.f64 a (*.f64 b c)) c) i) < 5e302

   1. Initial program 0.4

      \[2 \cdot \left(\left(x \cdot y + z \cdot t\right) - \left(\left(a + b \cdot c\right) \cdot c\right) \cdot i\right) \]

   2. Simplified 0.9

      \[\leadsto \color{blue}{2 \cdot \mathsf{fma}\left(z, t, \mathsf{fma}\left(\mathsf{fma}\left(b, c, a\right), i \cdot \left(-c\right), x \cdot y\right)\right)} \]

   3. Taylor expanded in z around 0 4.7

      \[\leadsto 2 \cdot \color{blue}{\left(y \cdot x + \left(t \cdot z + -1 \cdot \left(c \cdot \left(i \cdot \left(c \cdot b + a\right)\right)\right)\right)\right)} \]

   4. Simplified 4.7

      \[\leadsto 2 \cdot \color{blue}{\mathsf{fma}\left(y, x, \mathsf{fma}\left(i \cdot \mathsf{fma}\left(c, b, a\right), -c, t \cdot z\right)\right)} \]

   5. Taylor expanded in i around 0 4.7

      \[\leadsto 2 \cdot \mathsf{fma}\left(y, x, \color{blue}{-1 \cdot \left(c \cdot \left(i \cdot \left(c \cdot b + a\right)\right)\right) + t \cdot z}\right) \]

   6. Simplified 0.4

      \[\leadsto 2 \cdot \mathsf{fma}\left(y, x, \color{blue}{\mathsf{fma}\left(i, -c \cdot \mathsf{fma}\left(c, b, a\right), t \cdot z\right)}\right) \]

   if 5e302 < (*.f64 (*.f64 (+.f64 a (*.f64 b c)) c) i)

   1. Initial program 61.3

      \[2 \cdot \left(\left(x \cdot y + z \cdot t\right) - \left(\left(a + b \cdot c\right) \cdot c\right) \cdot i\right) \]

   2. Simplified 8.7

      \[\leadsto \color{blue}{2 \cdot \mathsf{fma}\left(z, t, \mathsf{fma}\left(\mathsf{fma}\left(b, c, a\right), i \cdot \left(-c\right), x \cdot y\right)\right)} \]

   3. Taylor expanded in z around 0 9.6

      \[\leadsto 2 \cdot \color{blue}{\left(y \cdot x + \left(t \cdot z + -1 \cdot \left(c \cdot \left(i \cdot \left(c \cdot b + a\right)\right)\right)\right)\right)} \]

   4. Simplified 9.5

      \[\leadsto 2 \cdot \color{blue}{\mathsf{fma}\left(y, x, \mathsf{fma}\left(i \cdot \mathsf{fma}\left(c, b, a\right), -c, t \cdot z\right)\right)} \]

   5. Taylor expanded in c around 0 6.2

      \[\leadsto 2 \cdot \mathsf{fma}\left(y, x, \mathsf{fma}\left(\color{blue}{c \cdot \left(i \cdot b\right) + i \cdot a}, -c, t \cdot z\right)\right) \]

   6. Applied egg-rr 2.0

      \[\leadsto 2 \cdot \mathsf{fma}\left(y, x, \mathsf{fma}\left(\color{blue}{\mathsf{fma}\left(c \cdot i, b, i \cdot a\right)}, -c, t \cdot z\right)\right) \]
3. Recombined 3 regimes into one program.

4. Final simplification 0.7

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(c \cdot \left(a + b \cdot c\right)\right) \cdot i \leq -\infty:\\ \;\;\;\;2 \cdot \mathsf{fma}\left(y, x, \mathsf{fma}\left(c \cdot \left(b \cdot i\right) + a \cdot i, -c, t \cdot z\right)\right)\\ \mathbf{elif}\;\left(c \cdot \left(a + b \cdot c\right)\right) \cdot i \leq 5 \cdot 10^{+302}:\\ \;\;\;\;2 \cdot \mathsf{fma}\left(y, x, \mathsf{fma}\left(i, c \cdot \left(-\mathsf{fma}\left(c, b, a\right)\right), t \cdot z\right)\right)\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \mathsf{fma}\left(y, x, \mathsf{fma}\left(\mathsf{fma}\left(c \cdot i, b, a \cdot i\right), -c, t \cdot z\right)\right)\\ \end{array} \]

Reproduce

herbie shell --seed 2022211 
(FPCore (x y z t a b c i)
  :name "Diagrams.ThreeD.Shapes:frustum from diagrams-lib-1.3.0.3, A"
  :precision binary64

  :herbie-target
  (* 2.0 (- (+ (* x y) (* z t)) (* (+ a (* b c)) (* c i))))

  (* 2.0 (- (+ (* x y) (* z t)) (* (* (+ a (* b c)) c) i))))